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 A266787 Binary representation of the n-th iteration of the "Rule 61" elementary cellular automaton starting with a single ON (black) cell. 2
 1, 11, 1100, 1010111, 1111000, 11010001111, 1111010000, 111010001111111, 1111010000000, 1111010001111111111, 1111010000000000, 11111010001111111111111, 1111010000000000000, 111111010001111111111111111, 1111010000000000000000, 1111111010001111111111111111111 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 LINKS Robert Price, Table of n, a(n) for n = 0..1000 Eric Weisstein's World of Mathematics, Elementary Cellular Automaton Stephen Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 55. Index entries for linear recurrences with constant coefficients, signature (0,11001,0,-10011000,0,10000000). FORMULA Conjectures from Colin Barker, Jan 03 2016 and Apr 18 2019: (Start) a(n) = 11001*a(n-2)-10011000*a(n-4)+10000000*a(n-6) for n>10. G.f.: (1 +11*x -9901*x^2 +889100*x^3 -979100*x^4 +7891000*x^5 -109001000*x^6 +1090110000*x^7 +9990000*x^8 -1100000000*x^9 +100000000*x^10) / ((1 -x)*(1 +x)*(1 -100*x)*(1 +100*x)*(1 -1000*x^2)). (End) From Karl V. Keller, Jr., Oct 20 2021 : (Start) The above conjectures are correct. a(n) = (10*100^n - 90999*1000^floor(n/2)/100 - 1)/9 for odd n > 4; a(n) = 111101*1000^(n/2)/100000 for even n > 4. (End) MATHEMATICA rule=61; rows=20; ca=CellularAutomaton[rule, {{1}, 0}, rows-1, {All, All}]; (* Start with single black cell *) catri=Table[Take[ca[[k]], {rows-k+1, rows+k-1}], {k, 1, rows}]; (* Truncated list of each row *) Table[FromDigits[catri[[k]]], {k, 1, rows}]   (* Binary Representation of Rows *) PROG (Python) print([1, 11, 1100, 1010111, 1111000] + [10*100**n - 90999*1000**(n//2)//100 - 1)//9 if n%2 else 111101*1000**(n//2)//100000 for n in range(5, 50)]) # Karl V. Keller, Jr., Oct 20 2021 CROSSREFS Cf. A266786, A266788. Sequence in context: A319424 A348269 A153435 * A109227 A133342 A110574 Adjacent sequences:  A266784 A266785 A266786 * A266788 A266789 A266790 KEYWORD nonn,easy AUTHOR Robert Price, Jan 03 2016 STATUS approved

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Last modified January 22 23:50 EST 2022. Contains 350504 sequences. (Running on oeis4.)